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Uniform asymptotics for S- and MM-regression estimators

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Abstract

In this paper we find verifiable regularity conditions to ensure that S-estimators of scale and regression and MM-estimators of regression are uniformly consistent and uniformly asymptotically normally distributed over contamination neighbourhoods. Moreover, we show how to calculate the size of these neighbourhoods. In particular, we find that, for MM-estimators computed with Tukey’s family of bisquare score functions, there is a trade-off between the size of these neighbourhoods and both the breakdown point of the S-estimators and the leverage of the contamination that is allowed in the neighbourhood. These results extend previous work of Salibian-Barrera and Zamar for location-scale to the linear regression model.

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Author information

Authors and Affiliations

  1. Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic

    Marek Omelka

  2. Jaroslav Hájek Centre for Theoretical and Applied Statistics, Charles University, Prague, Czech Republic

    Marek Omelka

  3. Department of Statistics, The University of British Columbia, Vancouver, BC, Canada

    Matías Salibián-Barrera

Authors
  1. Marek Omelka
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  2. Matías Salibián-Barrera
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Corresponding author

Correspondence to Matías Salibián-Barrera.

Additional information

M. Omelka’s research was supported by Research Project LC06024.

M. Salibián-Barrera’s research was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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Omelka, M., Salibián-Barrera, M. Uniform asymptotics for S- and MM-regression estimators. Ann Inst Stat Math 62, 897–927 (2010). https://doi.org/10.1007/s10463-008-0189-x

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  • DOI: https://doi.org/10.1007/s10463-008-0189-x

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